Non-collapsing volume estimate for local Kähler metrics in big cohomology classes

This paper develops a unified cohomological framework for analyzing n-derivations and their induced derivational automorphisms on nest algebras, extending classical Hochschild cohomology into the realm of higher-order operator algebraic structures. By constructing explicit n-cochain complexes adapted to the triangular nature of nest algebras, we classify n-derivations up to cohomological equivalence and provide precise criteria for their innerness. We demonstrate that the vanishing of higher-order cohomology groups corresponds to structural rigidity, while non-trivial cohomology classes reflect obstructions to decomposability and inner implementation. Moreover, we show that cohomologically trivial n-derivations preserve essential algebraic features, including the radical, center, invariant subspaces, and two-sided ideals. Through explicit computations of Hn(A,A) for low-dimensional examples, we verify the existence of non-trivial cohomological classes. Additionally, we introduce a dual cohomology theory for n-automorphisms via exponential mappings and establish a bidirectional correspondence between infinitesimal derivations and global automorphisms. This approach unifies derivational and automorphic symmetries under a single cohomological classification, offering new perspectives toward deformation theory, categorical dualities, and quantum operator structures.

Linear vector fields on Lie groups constitute a fundamental class of dynamical systems, as their flows are one-parameter subgroups of automorphisms and their infinitesimal behavior is entirely determined by derivations of the Lie algebra. When a Lie group is endowed with a Hamiltonian-type geometric structure, a natural problem is to determine whether such linear dynamics admit a global variational realization, and how such realizations can be interpreted in terms of reduced models of fluid motion. In the even-dimensional case, where the Lie group carries a symplectic structure, we establish a complete global criterion for the existence of Hamiltonians generating linear symplectic vector fields. The problem reduces to a single global obstruction: the de Rham cohomology class of the 1-form ιXω. Thus, every linear symplectic vector field on a simply connected Lie group is globally Hamiltonian, and when the obstruction vanishes, we provide an explicit constructive procedure to recover the Hamiltonian. On the affine group Aff+(1), this yields a fully explicit, finite-dimensional Hamiltonian model of a 1D ideal fluid with affine symmetries. We then treat odd-dimensional Lie groups, where symplectic geometry is unavailable. Using contact geometry as the canonical replacement, we prove a Hamiltonian lifting theorem ensuring the existence and uniqueness of the associated dynamics. The Reeb vector field appears as a distinguished vertical direction resolving the ambiguities of degenerate Hamiltonian systems. On the Heisenberg group H3, this gives a fully explicit contact Hamiltonian model of an effective non-conservative fluid mode. Finally, we interpret symplectic and contact theories within a unified geometric framework and discuss their relevance to geometric formulations of ideal (symplectic) and effective (contact) fluid equations on Lie groups.

We show that p-adic families of cohomology classes associated to symmetric spaces vary p-adically over small discs in weight space, without any ordinarity assumption. This generalises previous work of Loeffler, Zerbes and the author. Furthermore, we show that these families exhibit full variation in the cyclotomic direction, generalising previous constructions of Euler systems and p-adic L-functions. As an application we show that the Lemma–Flach Euler system of Loeffler–Skinner–Zerbes interpolates in Coleman families.

Higher-form symmetry in a tensor product Hilbert space is always emergent: The symmetry generators become genuinely topological only when the Gauss law is energetically enforced at low energies. In this Letter, we present a general method for defining the 't Hooft anomaly of higher-form symmetries in lattice models built on a tensor product Hilbert space. In (2+1)D, for given Gauss-law operators realized by finite-depth circuits that generate a finite 1-form G symmetry, we construct an index representing a cohomology class in H^{4}(B^{2}G,U(1)), which characterizes the corresponding 't Hooft anomaly. This construction generalizes the Else-Nayak characterization of 0-form symmetry anomalies. More broadly, under the assumption of a specified formulation of the p-form G symmetry action and Hilbert space structure in arbitrary d spatial dimensions, we show how to characterize the 't Hooft anomaly of the symmetry action by an index valued in H^{d+2}(B^{p+1}G,U(1)).

We compute the $\rm{SO}(n+1)$-equivariant mod $2$ Borel cohomology of the free iterated loop space $Z^{S^n}$ when $Z$ is a mod $2$ generalized Eilenberg Mac Lane space. When $n=1$, this recovers Bökstedt and Ottosen's computation for the free loop space. The highlight of our computation is a construction of cohomology classes using an $\mathrm{O}(n)$-equivariant evaluation map and a pushforward map.

Abstract We define a class of amenable Weyl group elements in the Lie types B, C, and D, which we propose as the analogs of vexillary permutations in these Lie types. Our amenable signed permutations index flagged theta and eta polynomials, which generalize the double theta and eta polynomials of Wilson and the author. In geometry, we obtain corresponding formulas for the cohomology classes of symplectic and orthogonal degeneracy loci.

In this study, we propose the idea of crossed homomorphisms between Lie–Yamaguti superalgebras and develop the Yamaguti cohomology theory of crossed homomorphisms. In light of this, we characterize linear deformations of crossing homomorphisms between Lie–Yamaguti superalgebras using this cohomology. We demonstrate that if two linear or formal deformations of a crossing homomorphism are similar, then their infinitesimals are in the same cohomology class in the first cohomology group. In addition, we show that an order n deformation of a crossing homomorphism can be extended to an order n+1 deformation if and only if the obstruction class in the second cohomology group is trivial.

Essential dimension of cohomology classes via valuation theory

Our aim is to determine the tautological algebra generated by the cohomology classes of the Brill-Noether loci in the rational cohomology of the moduli stack U C ( n , d ) of semistable bundles of rank n and degree d. We show that for a general smooth projective curve C of genus g ≥ 2 , d = 2 g − 2 , the tautological algebra of U C ( 2 , 2 g − 2 ) (resp. the moduli stack S U C ( 2 , L ) of semistable bundles of rank 2 and determinant L with deg ( L ) = 2 g − 2 ) is generated by the divisor classes (resp. the class of the Theta divisor Θ ). This is previously known in rank one situation, called the (classical) Porteous formula.

Starting from an abelian group G G and a factorizable ribbon Hopf G G -bialgebra H H , we construct a Topological Quantum Field Theory (TQFT) J H J_H for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in G G . When restricted to the subcategory of cobordisms with trivial decorations, our functor recovers a special case of Kerler–Lyubashenko TQFTs, namely those associated with factorizable ribbon Hopf algebras. Our result is inspired by the work of Blanchet–Costantino–Geer–Patureau, who constructed non-semisimple TQFTs for admissible decorated cobordisms using the unrolled quantum group of s l 2 \mathfrak {sl}_2 , and by that of Geer–Ha–Patureau, who reformulated the underlying invariants of admissible decorated 3 3 -manifolds using ribbon Hopf G G -coalgebras. Our work represents the first step towards a homological model for non-semisimple TQFTs decorated with cohomology classes that appears in a conjecture by the first two authors.

We introduce the notion of crossed homomorphisms between Lie-Yamaguti algebras and establish the cohomology theory of crossed homomorphisms via the Yamaguti cohomology. Accordingly, we use this cohomology to characterize linear deformations of crossed homomorphisms between Lie-Yamaguti algebras. If two linear or formal deformations of a crossed homomorphism are equivalent, then we show that their infinitesimals are in the same cohomology class. Moreover, we show that an order [Formula: see text] deformation of a crossed homomorphism can be extended to an order [Formula: see text] deformation if and only if the obstruction class is trivial.

Abstract For a connected reductive group $G$ over a local or global field $K$, we define a diamond (or power) operation $$ \begin{align*} &(\xi,n)\mapsto \xi^{\Diamond n}\,\colon\, \mathrm{H}^1\kern -0.8pt(K,G)\times{\mathbb Z}\to \mathrm{H}^1\kern -0.8pt(K,G)\end{align*} $$ of raising to power $n$ in the Galois cohomology pointed set. This operation is new when $K$ is a number field. We show that this power operation has many good properties. When $G$ is a torus, the set $\mathrm{H}^{1}\kern -0.8pt(K,G)$ has a natural group structure, and $\xi ^{\Diamond n}$ then coincides with the $n$-th power of $\xi $ in this group. On the other hand, we show that a power operation on $\mathrm{H}^{1}\kern -0.8pt(K,G)$, functorial in $G$, which we define over local and global fields, cannot be defined for an arbitrary field $K$. Our proof of this assertion relies on the results of Appendix B written by Philippe Gille. Using the power operation, for a cohomology class $\xi $ in $\mathrm{H}^{1}\kern -0.8pt(K,G)$ over local or global field, we define the period $\operatorname{per}(\xi )$ to be the least integer $n\geqslant 1$ such that $\xi ^{\Diamond n}=1$. We define the index $\operatorname{ind}(\xi )$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $\xi $. The period and index of a cohomology class generalize the period and index a central simple algebra over $K$. For any connected reductive group $G$ over a local or global field $K$, we show that $\operatorname{per}(\xi )$ divides $\operatorname{ind}(\xi )$ and that $\operatorname{ind}(\xi )$ may be strictly greater than $\operatorname{per}(\xi )$, but they always have the same prime factors.

We study resonant differential forms at zero for transitive Anosov flows on 3-manifolds.We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure.Such flows have two distinguished Sinai-Ruelle-Bowen 3-forms, RB , and the cohomology classes OE X RB (where X is the infinitesimal generator of the flow) play a key role in the determination of the space of resonant 1-forms.When both classes vanish we associate to the flow a helicity that naturally extends the classical notion associated with null-homologous volume-preserving flows.We provide a general theory that includes horocyclic invariance of resonant 1-forms and SRB-measures as well as the local geometry of the maps X 7 !OE X RB near a null-homologous volume-preserving flow.Next, we study several relevant classes of examples.Among these are thermostats associated with holomorphic quadratic differentials, giving rise to quasi-Fuchsian flows as introduced by Ghys (1992).For these flows we compute explicitly all resonant 1-forms at zero, we show that OE X RB D 0 and give an explicit formula for the helicity.In addition we show that a generic time change of a quasi-Fuchsian flow is semisimple and thus the order of vanishing of the Ruelle zeta function at zero is .M /, the same as in the geodesic flow case.In contrast, we show that if .M; g/ is a closed surface of negative curvature, the Gaussian thermostat driven by a (small) harmonic 1-form has a Ruelle zeta function whose order of vanishing at zero is .M / 1.

ABSTRACT For an algebraic torus defined over a local (or global) field F, a celebrated result of R.P. Langlands establishes a natural homomorphism from the group of continuous cohomology classes of the Weil group, valued in the dual torus, onto the space of complex characters of the rational points of the torus (or automorphic characters in the global case). We expand on this result by detailing its topological aspects. We show that if we topologize the relevant spaces of continuous homomorphisms and continuous cochains using the compact-open topology, Langlands’s map becomes a (surjective, finite-to-one) homomorphism of abelian complex Lie groups. Moreover, we demonstrate that, in both the local and global settings, the subset of unramified characters is the identity component of the relevant space of characters. Finally, we compare the group of unramified characters with the Galois (co)invariants of the dual torus.

Abstract Let be the moduli space of algebraic curves of genus with boundaries and marked points, and its compactly supported cohomology group. We prove that the collection of ‐modules has the structure of a properad (called the gravity properad) such that it contains the Getzler's gravity operad as the sub‐collection . The properadic structure in is highly nontrivial and generates higher genus cohomology classes from lower ones (which is demonstrated on infinitely many nontrivial examples producing higher genus cohomology classes from just zero genus ones). Moreover, we prove that the generators of the 1‐dimensional cohomology groups , and satisfy with respect to this properadic structure the relations of the (degree shifted) quasi‐Lie bialgebra, a fact making the totality of cohomology groups into a complex with the differential fully determined by the just mentioned three cohomology classes. It is proven that this complex contains infinitely many nontrivial cohomology classes, all coming from Kontsevich's odd graph complex. The prop structure in is established with the help of Willwacher's twisting endofunctor (in the category of properads under the operad of Lie algebras) and Costello's theory of moduli spaces of nodal disks with marked boundaries and internal marked points.

Abstract This is a continuation of our previous work on quantitative stability for complex Monge-Ampère equation. In the recent paper [21], we treated the stability question for fixed cohomology classes and fixed prescribed singularity types. In this work, we establish quantitative stability estimates for complex Monge-Ampère equations when both the cohomology class and the prescribed singularity vary.

Abstract We prove a Painlevé theorem for bounded quasiregular curves in Euclidean spaces extending removability results for quasiregular mappings due to Iwaniec and Martin. The theorem is proved by extending a fundamental inequality for volume forms to calibrations and proving a Caccioppoli inequality for quasiregular curves. We also establish a qualitatively sharp removability theorem for quasiregular curves whose target is a Riemannian manifold with sectional curvature bounded from above and injectivity radius bounded from below. As an application, we extend a theorem of Bonk and Heinonen for quasiregular mappings to the setting of quasiregular curves: every non-constant quasiregular $$\omega $$ ω -curve from $$\mathbb {R}^n$$ R n into $$( N, \omega )$$ ( N , ω ) , where the bounded cohomology class of $$\omega $$ ω is in the bounded Künneth ideal, has infinite energy.

Abstract We prove a correspondence, for Riemannian manifolds with self-dual Weyl tensor, between twistor functions and solutions to the Teukolsky equations for any conformal and spin weights. In particular, we give a contour integral formula for solutions to the Teukolsky equations, and we find a recursion operator that generates an infinite family of solutions and leads to the construction of Čech representatives and sheaf cohomology classes on twistor space. Apart from the general conformally self-dual case, examples include self-dual black holes, scalar-flat Kähler surfaces, and quaternionic-Kähler metrics, where we map the Teukolsky equation to the conformal wave equation, establish new relations to the linearised Przanowski equation, and find new classes of quaternionic deformations.

We introduce new algebraic structures associated with heptagon relations—higher analogue of the well-known pentagon. The main points we deal with are: (i) polygon relations as algebraic imitations of Pachner moves, on the example of heptagon, (ii) parameterization of heptagon relations by simplicial 3-cocycles, (iii) applications to invariants of pairs (piecewise linear 5-manifold, a third cohomology class on it).